Properties

Label 3549.2.a.ba
Level $3549$
Weight $2$
Character orbit 3549.a
Self dual yes
Analytic conductor $28.339$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 13 x^{6} + 22 x^{5} + 57 x^{4} - 72 x^{3} - 96 x^{2} + 64 x + 61\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} - q^{3} + ( 1 + \beta_{1} + \beta_{4} + \beta_{5} ) q^{4} + ( -1 + \beta_{2} + \beta_{5} ) q^{5} + \beta_{1} q^{6} + q^{7} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} - q^{3} + ( 1 + \beta_{1} + \beta_{4} + \beta_{5} ) q^{4} + ( -1 + \beta_{2} + \beta_{5} ) q^{5} + \beta_{1} q^{6} + q^{7} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{8} + q^{9} + ( 1 + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{10} + ( -2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{11} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} ) q^{12} -\beta_{1} q^{14} + ( 1 - \beta_{2} - \beta_{5} ) q^{15} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{16} + ( 1 + \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{17} -\beta_{1} q^{18} + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{7} ) q^{19} + ( \beta_{2} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{20} - q^{21} + ( 1 + 4 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{22} + ( -\beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} ) q^{23} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} ) q^{24} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{25} - q^{27} + ( 1 + \beta_{1} + \beta_{4} + \beta_{5} ) q^{28} + ( 2 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{29} + ( -1 - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{30} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{31} + ( -1 - 3 \beta_{1} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{32} + ( 2 + \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{33} + ( -3 - \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{34} + ( -1 + \beta_{2} + \beta_{5} ) q^{35} + ( 1 + \beta_{1} + \beta_{4} + \beta_{5} ) q^{36} + ( \beta_{2} + 2 \beta_{6} - \beta_{7} ) q^{37} + ( -1 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} - 3 \beta_{7} ) q^{38} + ( -\beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} - 3 \beta_{5} + 2 \beta_{7} ) q^{40} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{41} + \beta_{1} q^{42} + ( 3 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{43} + ( -7 - 4 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{44} + ( -1 + \beta_{2} + \beta_{5} ) q^{45} + ( 2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{46} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + 3 \beta_{7} ) q^{47} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{48} + q^{49} + ( -2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 5 \beta_{5} + \beta_{6} + \beta_{7} ) q^{50} + ( -1 - \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{51} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - \beta_{7} ) q^{53} + \beta_{1} q^{54} + ( 2 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{55} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{56} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{57} + ( -2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 5 \beta_{7} ) q^{58} + ( -6 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{59} + ( -\beta_{2} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{60} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{61} + ( -5 + \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{62} + q^{63} + ( 4 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{64} + ( -1 - 4 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{66} + ( 3 - 2 \beta_{1} + \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{67} + ( -4 + 5 \beta_{1} + \beta_{3} - 4 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{68} + ( \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} ) q^{69} + ( 1 + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{70} + ( -4 - 3 \beta_{2} - \beta_{3} - \beta_{6} ) q^{71} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{72} + ( 1 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{7} ) q^{73} + ( -\beta_{1} - 3 \beta_{2} - 6 \beta_{3} - 2 \beta_{5} ) q^{74} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{75} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} ) q^{76} + ( -2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{77} + ( 5 - 3 \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{7} ) q^{79} + ( 1 + \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{80} + q^{81} + ( -4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{82} + ( -7 + 2 \beta_{1} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{83} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} ) q^{84} + ( -2 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{85} + ( -9 - 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{86} + ( -2 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{87} + ( 3 + 8 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} ) q^{88} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 4 \beta_{7} ) q^{89} + ( 1 + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{90} + ( -4 - 4 \beta_{1} - 4 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{92} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{93} + ( 3 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - 3 \beta_{7} ) q^{94} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{95} + ( 1 + 3 \beta_{1} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{96} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{97} -\beta_{1} q^{98} + ( -2 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{2} - 8q^{3} + 14q^{4} - 6q^{5} + 2q^{6} + 8q^{7} - 12q^{8} + 8q^{9} + O(q^{10}) \) \( 8q - 2q^{2} - 8q^{3} + 14q^{4} - 6q^{5} + 2q^{6} + 8q^{7} - 12q^{8} + 8q^{9} - 4q^{10} - 16q^{11} - 14q^{12} - 2q^{14} + 6q^{15} + 10q^{16} - 2q^{17} - 2q^{18} - 14q^{19} + 10q^{20} - 8q^{21} + 18q^{22} - 6q^{23} + 12q^{24} + 10q^{25} - 8q^{27} + 14q^{28} + 12q^{29} + 4q^{30} - 16q^{31} - 26q^{32} + 16q^{33} - 32q^{34} - 6q^{35} + 14q^{36} + 8q^{37} + 12q^{38} - 14q^{40} + 4q^{41} + 2q^{42} + 14q^{43} - 68q^{44} - 6q^{45} + 28q^{46} - 6q^{47} - 10q^{48} + 8q^{49} - 32q^{50} + 2q^{51} + 14q^{53} + 2q^{54} - 2q^{55} - 12q^{56} + 14q^{57} - 24q^{58} - 50q^{59} - 10q^{60} - 2q^{61} - 20q^{62} + 8q^{63} + 24q^{64} - 18q^{66} + 16q^{67} - 36q^{68} + 6q^{69} - 4q^{70} - 34q^{71} - 12q^{72} + 8q^{73} - 6q^{74} - 10q^{75} + 4q^{76} - 16q^{77} + 46q^{79} + 24q^{80} + 8q^{81} - 42q^{82} - 42q^{83} - 14q^{84} - 20q^{85} - 50q^{86} - 12q^{87} + 62q^{88} - 28q^{89} - 4q^{90} - 58q^{92} + 16q^{93} + 24q^{94} - 24q^{95} + 26q^{96} - 16q^{97} - 2q^{98} - 16q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 13 x^{6} + 22 x^{5} + 57 x^{4} - 72 x^{3} - 96 x^{2} + 64 x + 61\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{7} + 3 \nu^{6} + 21 \nu^{5} - 27 \nu^{4} - 56 \nu^{3} + 64 \nu^{2} + 16 \nu - 29 \)\()/13\)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 7 \nu^{2} + 4 \nu + 8 \)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{7} - 3 \nu^{6} - 21 \nu^{5} + 27 \nu^{4} + 69 \nu^{3} - 64 \nu^{2} - 81 \nu + 16 \)\()/13\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{7} + 3 \nu^{6} + 21 \nu^{5} - 27 \nu^{4} - 69 \nu^{3} + 77 \nu^{2} + 68 \nu - 55 \)\()/13\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{7} + 3 \nu^{6} + 34 \nu^{5} - 40 \nu^{4} - 160 \nu^{3} + 129 \nu^{2} + 172 \nu - 55 \)\()/13\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 5 \nu^{6} - 17 \nu^{5} - 58 \nu^{4} + 80 \nu^{3} + 202 \nu^{2} - 112 \nu - 200 \)\()/13\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(7 \beta_{5} + 8 \beta_{4} + \beta_{3} + \beta_{2} + 8 \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(\beta_{6} + 2 \beta_{5} + 11 \beta_{4} + \beta_{3} + 8 \beta_{2} + 31 \beta_{1} + 9\)
\(\nu^{6}\)\(=\)\(2 \beta_{7} + \beta_{6} + 43 \beta_{5} + 55 \beta_{4} + 12 \beta_{3} + 12 \beta_{2} + 59 \beta_{1} + 80\)
\(\nu^{7}\)\(=\)\(3 \beta_{7} + 12 \beta_{6} + 23 \beta_{5} + 94 \beta_{4} + 15 \beta_{3} + 54 \beta_{2} + 206 \beta_{1} + 79\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.72116
2.54225
1.67113
1.32829
−0.756173
−1.06752
−2.14586
−2.29328
−2.72116 −1.00000 5.40472 −1.58278 2.72116 1.00000 −9.26480 1.00000 4.30699
1.2 −2.54225 −1.00000 4.46304 4.07309 2.54225 1.00000 −6.26168 1.00000 −10.3548
1.3 −1.67113 −1.00000 0.792659 −2.68351 1.67113 1.00000 2.01762 1.00000 4.48449
1.4 −1.32829 −1.00000 −0.235645 −1.55828 1.32829 1.00000 2.96959 1.00000 2.06984
1.5 0.756173 −1.00000 −1.42820 −4.01537 −0.756173 1.00000 −2.59231 1.00000 −3.03631
1.6 1.06752 −1.00000 −0.860399 0.994065 −1.06752 1.00000 −3.05354 1.00000 1.06118
1.7 2.14586 −1.00000 2.60470 −1.91954 −2.14586 1.00000 1.29759 1.00000 −4.11906
1.8 2.29328 −1.00000 3.25913 0.692320 −2.29328 1.00000 2.88753 1.00000 1.58768
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3549.2.a.ba 8
13.b even 2 1 3549.2.a.bc 8
13.f odd 12 2 273.2.bd.b 16
39.k even 12 2 819.2.ct.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bd.b 16 13.f odd 12 2
819.2.ct.c 16 39.k even 12 2
3549.2.a.ba 8 1.a even 1 1 trivial
3549.2.a.bc 8 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\):

\(T_{2}^{8} + \cdots\)
\(T_{5}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 61 - 64 T - 96 T^{2} + 72 T^{3} + 57 T^{4} - 22 T^{5} - 13 T^{6} + 2 T^{7} + T^{8} \)
$3$ \( ( 1 + T )^{8} \)
$5$ \( -143 + 40 T + 312 T^{2} + 70 T^{3} - 177 T^{4} - 104 T^{5} - 7 T^{6} + 6 T^{7} + T^{8} \)
$7$ \( ( -1 + T )^{8} \)
$11$ \( -764 + 2064 T + 4457 T^{2} - 1352 T^{3} - 1647 T^{4} - 218 T^{5} + 60 T^{6} + 16 T^{7} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( 15433 + 10490 T - 14550 T^{2} - 1668 T^{3} + 2421 T^{4} - 46 T^{5} - 91 T^{6} + 2 T^{7} + T^{8} \)
$19$ \( -3068 + 2400 T + 3861 T^{2} - 1070 T^{3} - 1829 T^{4} - 432 T^{5} + 19 T^{6} + 14 T^{7} + T^{8} \)
$23$ \( -4784 + 1880 T + 7297 T^{2} + 2394 T^{3} - 890 T^{4} - 530 T^{5} - 57 T^{6} + 6 T^{7} + T^{8} \)
$29$ \( -285503 + 155842 T + 77709 T^{2} - 48320 T^{3} + 1605 T^{4} + 1492 T^{5} - 106 T^{6} - 12 T^{7} + T^{8} \)
$31$ \( 39232 + 42976 T - 1435 T^{2} - 17348 T^{3} - 7644 T^{4} - 1134 T^{5} + 9 T^{6} + 16 T^{7} + T^{8} \)
$37$ \( -472811 + 287376 T + 16646 T^{2} - 31196 T^{3} + 2211 T^{4} + 964 T^{5} - 105 T^{6} - 8 T^{7} + T^{8} \)
$41$ \( -85184 + 286272 T - 200512 T^{2} - 30256 T^{3} + 11724 T^{4} + 656 T^{5} - 195 T^{6} - 4 T^{7} + T^{8} \)
$43$ \( 774292 - 1281620 T + 607653 T^{2} - 92716 T^{3} - 5638 T^{4} + 2618 T^{5} - 123 T^{6} - 14 T^{7} + T^{8} \)
$47$ \( -41036 - 77872 T + 133341 T^{2} + 105890 T^{3} + 15798 T^{4} - 1594 T^{5} - 265 T^{6} + 6 T^{7} + T^{8} \)
$53$ \( -8807 + 7238 T + 28748 T^{2} - 17386 T^{3} - 6458 T^{4} + 2434 T^{5} - 124 T^{6} - 14 T^{7} + T^{8} \)
$59$ \( -1521692 - 931512 T + 53489 T^{2} + 186428 T^{3} + 65628 T^{4} + 11120 T^{5} + 1029 T^{6} + 50 T^{7} + T^{8} \)
$61$ \( 338917 - 140026 T - 99144 T^{2} + 9654 T^{3} + 6138 T^{4} - 238 T^{5} - 136 T^{6} + 2 T^{7} + T^{8} \)
$67$ \( -345392 - 697504 T + 379361 T^{2} - 13576 T^{3} - 20298 T^{4} + 3684 T^{5} - 123 T^{6} - 16 T^{7} + T^{8} \)
$71$ \( 25924 + 100584 T - 40279 T^{2} - 68954 T^{3} - 17025 T^{4} - 104 T^{5} + 336 T^{6} + 34 T^{7} + T^{8} \)
$73$ \( 97837 - 122228 T - 511270 T^{2} - 49718 T^{3} + 19617 T^{4} + 1314 T^{5} - 243 T^{6} - 8 T^{7} + T^{8} \)
$79$ \( -7916 + 328532 T - 535107 T^{2} + 170512 T^{3} - 7678 T^{4} - 4238 T^{5} + 741 T^{6} - 46 T^{7} + T^{8} \)
$83$ \( -1196 - 1240 T + 19041 T^{2} - 19780 T^{3} - 465 T^{4} + 2942 T^{5} + 596 T^{6} + 42 T^{7} + T^{8} \)
$89$ \( -5457824 + 249576 T + 1087977 T^{2} + 153776 T^{3} - 21365 T^{4} - 4356 T^{5} + 19 T^{6} + 28 T^{7} + T^{8} \)
$97$ \( 208 + 3576 T + 12441 T^{2} + 11060 T^{3} - 752 T^{4} - 1782 T^{5} - 119 T^{6} + 16 T^{7} + T^{8} \)
show more
show less